Benefiting from well-trained deep neural networks (DNNs), model compression have captured special attention for computing resource limited equipment, especially edge devices. Knowledge distillation (KD) is one of the widely used compression techniques for edge deployment, by obtaining a lightweight student model from a well-trained teacher model released on public platforms. However, it has been empirically noticed that the backdoor in the teacher model will be transferred to the student model during the process of KD. Although numerous KD methods have been proposed, most of them focus on the distillation of a high-performing student model without robustness consideration. Besides, some research adopts KD techniques as effective backdoor mitigation tools, but they fail to perform model compression at the same time. Consequently, it is still an open problem to well achieve two objectives of robust KD, i.e., student model's performance and backdoor mitigation. To address these issues, we propose RobustKD, a robust knowledge distillation that compresses the model while mitigating backdoor based on feature variance. Specifically, RobustKD distinguishes the previous works in three key aspects: (1) effectiveness: by distilling the feature map of the teacher model after detoxification, the main task performance of the student model is comparable to that of the teacher model; (2) robustness: by reducing the characteristic variance between the teacher model and the student model, it mitigates the backdoor of the student model under backdoored teacher model scenario; (3) generic: RobustKD still has good performance in the face of multiple data models (e.g., WRN 28-4, Pyramid-200) and diverse DNNs (e.g., ResNet50, MobileNet).